Excercise 2.4 in Jurgen Jost's PDE “Harnack's inequality” for harmonic functions defined on a ball
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Let $u:B(0,R)subset mathbb{R^d}rightarrowmathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)leq u(x)leq dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$.
I know what the usual inequality says and I have the proof of that at hand, however I do not know how to begin. Thanks in advance for any hints.
inequality pde harmonic-functions
asked Jan 11 at 18:35
AlfdavAlfdav
977
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add a comment |
$begingroup$
Let $u:B(0,R)subset mathbb{R^d}rightarrowmathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)leq u(x)leq dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$.
I know what the usual inequality says and I have the proof of that at hand, however I do not know how to begin. Thanks in advance for any hints.
inequality pde harmonic-functions
asked Jan 11 at 18:35
AlfdavAlfdav
977
$endgroup$
1
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
1
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Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
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– Martin R
Jan 11 at 18:44
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I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47
add a comment |
$begingroup$
Let $u:B(0,R)subset mathbb{R^d}rightarrowmathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)leq u(x)leq dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$.
I know what the usual inequality says and I have the proof of that at hand, however I do not know how to begin. Thanks in advance for any hints.
inequality pde harmonic-functions
asked Jan 11 at 18:35
AlfdavAlfdav
977
$endgroup$
Let $u:B(0,R)subset mathbb{R^d}rightarrowmathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)leq u(x)leq dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$.
I know what the usual inequality says and I have the proof of that at hand, however I do not know how to begin. Thanks in advance for any hints.
inequality pde harmonic-functions
inequality pde harmonic-functions
asked Jan 11 at 18:35
AlfdavAlfdav
977
asked Jan 11 at 18:35
AlfdavAlfdav
977
asked Jan 11 at 18:35
AlfdavAlfdav
977
asked Jan 11 at 18:35
AlfdavAlfdav
977
asked Jan 11 at 18:35
AlfdavAlfdav
977
977
1
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
1
$begingroup$
Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
$endgroup$
– Martin R
Jan 11 at 18:44
$begingroup$
I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47
add a comment |
1
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
1
$begingroup$
Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
$endgroup$
– Martin R
Jan 11 at 18:44
$begingroup$
I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47
1
1
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
1
1
$begingroup$
Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
$endgroup$
– Martin R
Jan 11 at 18:44
$begingroup$
Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
$endgroup$
– Martin R
Jan 11 at 18:44
$begingroup$
I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47
add a comment |
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1
$begingroup$
Do you know the Poisson integral formula in this context? It must follow from that...
$endgroup$
– David C. Ullrich
Jan 11 at 18:42
1
$begingroup$
Isn't that exactly en.wikipedia.org/wiki/Harnack%27s_inequality#The_statement ?
$endgroup$
– Martin R
Jan 11 at 18:44
$begingroup$
I dont but, I was just reading from another book, "PDE by L. Evans" and it has the same excercise but gives the following hint "Use Poisson's formula for the ball to prove..." which is (I believe) what you are suggesting. I'll give it a try with said integral and let you know.
$endgroup$
– Alfdav
Jan 11 at 18:45
$begingroup$
thanks to both of you
$endgroup$
– Alfdav
Jan 12 at 20:47